Search

This paper explores the integration of Visual Code Assistants in Integrated Development Environments (IDEs). In Software Engineering, whiteboard sketching is often the initial step before coding, serving as a crucial collaboration tool for developers. Previous studies have investigated patterns in SE sketches and how they are used in practice, yet methods for directly using these sketches for code generation remain limited. The emergence of visually-equipped large language models presents an opportunity to bridge this gap, which is the focus of our research. In this paper, we built a first prototype of a Visual Code Assistant to get user feedback regarding in-IDE sketch-to-code tools. We conduct an experiment with 19 data scientists, most of whom regularly sketch as part of their job. We investigate developers' mental models by analyzing patterns commonly observed in their sketches when developing an ML workflow. Analysis indicates that diagrams were the preferred organizational component (52.6%), often accompanied by lists (42.1%) and numbered points (36.8%). Our tool converts their sketches into a Python notebook by querying an LLM. We use an LLM-as-judge setup to score the quality of the generated code, finding that even brief sketching can effectively generate useful code outlines. We also find a positive correlation between sketch time and the quality of the generated code. We conclude the study by conducting extensive interviews to assess the tool's usefulness, explore potential use cases, and understand developers' needs. As noted by participants, promising applications for these assistants include education, prototyping, and collaborative settings. Our findings signal promise for the next generation of Code Assistants to integrate visual information, both to improve code generation and to better leverage developers' existing sketching practices.

A systematic construction for supersymmetric negative (non-local) flows for mKdV and KdV based on $sl(2,1)$ with a principal gradation is proposed in this paper. We show that smKdV and sKdV can be mapped onto each other through a gauge super Miura transformation, together with an additional condition for the negative flows, which ensure the supersymmetry of the negative sKdV flow. In addition, we classify both smKdV and sKdV flows with respect to the vacuum (boundary) solutions accepted for each flow, whether zero or non-zero. Each vacuum is used to derive both soliton solutions and the Heisenberg subalgebra for the smKdV hierarchy, where we present novel vertex operators and solutions for non-zero bosonic and fermionic vacuum. Finally, the gauge Miura transformation is used to obtain the sKdV solutions, which exhibit a rich degeneracy due to both multiple gauge super Miura transformations and multiple vacuum possibilities., Comment: 49 pages, 2 figures

The partial information decomposition (PID) framework is concerned with decomposing the information that a set of (two or more) random variables (the sources) has about another variable (the target) into three types of information: unique, redundant, and synergistic. Classical information theory alone does not provide a unique way to decompose information in this manner and additional assumptions have to be made. One often overlooked way to achieve this decomposition is using a so-called measure of union information - which quantifies the information that is present in at least one of the sources - from which a synergy measure stems. In this paper, we introduce a new measure of union information based on adopting a communication channel perspective, compare it with existing measures, and study some of its properties. We also include a comprehensive critical review of characterizations of union information and synergy measures that have been proposed in the literature.

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat {\cal{G}} $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $\hat {sl}(2,1)$ super-algebra., Comment: 22 pages Latex

The construction of negative grade KdV hierarchy is proposed in terms of a Miura-gauge transformation. Such gauge transformation is employed within the zero curvature representation and maps the Lax operator of the mKdV into its couterpart within the KdV setting. Each odd negative KdV flow is obtained from an odd and its subsequent even negative mKdV flows. The negative KdV flows are shown to inherit the two different vacua structure that characterizes the associated mKdV flows., Comment: 8 pages, 34th International Colloquium on Group Theoretical Methods in Physics

The gauge-Miura correspondence establishes a map between the entire KdV and mKdV hierarchies, including positive and also negative flows, from which new relations besides the standard Miura transformation arise. We use this correspondence to classify solutions of the KdV hierarchy in terms of elementary tau functions of the mKdV hierarchy under both zero and nonzero vacua. We illustrate how interesting nonlinear phenomena can be described analytically from this construction, such as ``rogue waves'' of a complex KdV system that corresponds to a limit of a vector nonlinear Schr\"odinger equation., Comment: 8 pages. Int. Symp. Quantum Theory and Symmetries (QST12)

We present a construction of a class of rational solutions of the Painlev\'e V equation that exhibit a two-fold degeneracy, meaning that there exist two distinct solutions that share identical parameters. The fundamental object of our study is the orbit of translation operators of $A^{(1)}_{3}$ affine Weyl group acting on the underlying seed solution that only allows action of some symmetry operations. By linking points on this orbit to rational solutions, we establish conditions for such degeneracy to occur after involving in the construction additional B\"acklund transformations that are inexpressible as translation operators. This approach enables us to derive explicit expressions for these degenerate solutions. An advantage of this formalism is that it easily allows generalization to higher Painlev\'e systems associated with dressing chains of even period $N>4$., Comment: 15 pages

In this work, a multifractal framework is proposed to investigate the effects of current sheets in solar wind turbulence. By using multifractal detrended fluctuation analysis coupled with surrogate methods and volatility, two solar wind magnetic field time series are investigated, one with current sheets and one without current sheets. Despite the lack of extreme-events intermittent bursts in the current sheet-free series, both series are shown to be strongly multifractal, although the current sheet-free series displays an almost linear behavior for the scaling exponent of structure functions. Long-range correlations are shown to be the main source of multifractality for the series without current sheets, while a combination of heavy-tail distribution and nonlinear correlations are responsible for multifractality in the series with current sheets. The multifractality in both time series is formally shown to be associated with an energy-cascade process using the p-model., Comment: Accepted by MNRAS

Under special conditions the Painlev\'e V equation has more than one rational solution solving it with the same parameters. In the setting of formalism that identifies points on orbits of the fundamental shift operators of $A^{(1)}_{3}$ affine Weyl group with rational solutions we derive conditions for such non-uniqueness to occur. We identify the seed solutions from which the non-unique solutions are generated and put forward a method to systematically obtain their closed expressions from the underlying seed solutions., Comment: 20 pages

The partial information decomposition (PID) framework is concerned with decomposing the information that a set of random variables has with respect to a target variable into three types of components: redundant, synergistic, and unique. Classical information theory alone does not provide a unique way to decompose information in this manner and additional assumptions have to be made. Recently, Kolchinsky proposed a new general axiomatic approach to obtain measures of redundant information, based on choosing an order relation between information sources (equivalently, order between communication channels). In this paper, we exploit this approach to introduce three new measures of redundant information (and the resulting decompositions) based on well-known preorders between channels, thus contributing to the enrichment of the PID landscape. We relate the new decompositions to existing ones, study some of their properties, and provide examples illustrating their novelty. As a side result, we prove that any preorder that satisfies Kolchinsky's axioms yields a decomposition that meets the axioms originally introduced by Williams and Beer when they first propose the PID., Comment: 13 pages, 1 figure

We show how the zero-curvature equations based on a loop algebra of $D_4$ with a principal gradation reduce via self-similarity limit to a polynomial Hamiltonian system of coupled Painlev\'e III models with four canonical variables and $D_4^{(1)}$ affine Weyl group symmetry., Comment: 14 pages

The self-avoid random walk algorithm has been extensively used in the study of polymers. In this work we study the basic properties of the trajectories generated with this algorithm when two interactions are added to it: contact and folding interaction. These interactions represent the internal forces of the polymer as well as the effect of the solvent. When independently added to the algorithm, the contact interaction creates the compact phase while the folding one creates the extended phase. These are the consequences of the typical event of each interaction. On the other hand, when this typical event is avoided there is no established phase on the system. When simultaneously added, there is a competition between the interactions and the folding one is dominant over the contact one. The resulting phase is always the extended one with and without the contact interaction., Comment: 8 pages, 7 figures

The KdV hierarchy is a paradigmatic example of the rich mathematical structure underlying integrable systems and has far-reaching connections in several areas of theoretical physics. While the positive part of the KdV hierarchy is well known, in this paper we consider an affine Lie algebraic construction for its negative part. We show that the original Miura transformation can be extended to a gauge transformation that implies several new types of relations among the negative flows of the KdV and mKdV hierarchies. Contrary to the positive flows, such a ``gauge-Miura'' correspondence becomes degenerate whereby more than one negative mKdV model is mapped into a single negative KdV model. For instance, the sine-Gordon and another negative mKdV flow are mapped into a single negative KdV flow which inherits solutions of both former models. The gauge-Miura correspondence implies a rich degeneracy regarding solutions of these hierarchies. We obtain similar results for the generalized KdV and mKdV hierachies constructed with the affine Lie algebra $\widehat{s\ell}(r+1)$. In this case the first negative mKdV flow corresponds to an affine Toda field theory and the gauge-Miura correspondence yields its KdV counterpart. In particular, we show explicitly a KdV analog of the Tzitz\'{e}ica-Bullough-Dodd model. In short, we uncover a rich mathematical structure for the negative flows of integrable hierarchies obtaining novel relations and integrable systems., Comment: New results for commuting flows of mKdV and KdV hierarchies. Matches the version to appear on JHEP

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction identifies rational solutions with points on orbits of fundamental shift operators acting on first-order polynomial solutions derived for dressing chains of even periodicity. We also obtain conditions for the existence of special function solutions that occur for a special class of first-order polynomial solutions. For the special case of the N=4 dressing chain equations the method yields all the known rational solutions of Painlev\'e V equation. They are obtained through action of shift operators on the two independent first-order polynomial solutions. The formalism naturally extends to N=6 and beyond as shown in the paper., Comment: 29 pages

A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine $A_{2r}^{(2)}$ Lie algebra. The zero curvature representation of the time evolution equations are shown to be classified according to its algebraic structure and according to its vacuum solutions. It is shown that a class of models admit both zero and constant (non zero) vacuum solutions. Another, consists essentially of integral non-local equations and can be classified into two sub-classes, one admitting zero vacuum and another of constant, non zero vacuum solutions. The two dimensional gauge potentials in the vacuum plays a crucial ingredient and are shown to be expanded in powers of the vacuum parameter $v_0$. Soliton solutions are constructed from vertex operators, which for the non zero vacuum solutions, correspond to deformations characterized by $v_0$., Comment: Latex 24 pages

We propose a Hamiltonian formalism for $N$ periodic dressing chain with the even number $N$. The formalism is based on Dirac reduction applied to the $N+1$ periodic dressing chain with the odd number $N+1$ for which the Hamiltonian formalism is well known. The Hamilton dressing chain equations in the $N$ even case depend explicitly on a pair of conjugated Dirac constraints and are equivalent to $A^{(1)}_{N-1}$ invariant symmetric Painlev\'e equations., Comment: 13 pages, comment added in subsection 3.1

The construction of Miura and B\"acklund transformations for $A_n$ mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known $sl(2)$ case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miura-gauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of $A_n$ The construction of generalized gauge-B\"acklund transformation for the $A_n$-KdV hierarchy is obtained as a composition of Miura and B\"acklund-gauge transformations for $A_n$-mKdV hierarchy. The zero curvature representation provide a framework which is universal within all flows and generate systematically B\"acklund transformations for the entirely hierarchy., Comment: Latex 18 pages, corrections and improvements in the text

We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlev\'e equations. The $A^{(1)}_{N-1}$ B\"acklund symmetries of dressing equations and Painlev\'e equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.

Kernel pruning methods have been proposed to speed up, simplify, and improve explanation of convolutional neural network (CNN) models. However, the effectiveness of a simplified model is often below the original one. In this letter, we present new methods based on objective and subjective relevance criteria for kernel elimination in a layer-by-layer fashion. During the process, a CNN model is retrained only when the current layer is entirely simplified, by adjusting the weights from the next layer to the first one and preserving weights of subsequent layers not involved in the process. We call this strategy \emph{progressive retraining}, differently from kernel pruning methods that usually retrain the entire model after each simplification action -- e.g., the elimination of one or a few kernels. Our subjective relevance criterion exploits the ability of humans in recognizing visual patterns and improves the designer's understanding of the simplification process. The combination of suitable relevance criteria and progressive retraining shows that our methods can increase effectiveness with considerable model simplification. We also demonstrate that our methods can provide better results than two popular ones and another one from the state-of-the-art using four challenging image datasets., Comment: 7 pages, 4 figures. This paper was submitted to Pattern Recognition Letters

Intestinal parasites are responsible for several diseases in human beings. In order to eliminate the error-prone visual analysis of optical microscopy slides, we have investigated automated, fast, and low-cost systems for the diagnosis of human intestinal parasites. In this work, we present a hybrid approach that combines the opinion of two decision-making systems with complementary properties: ($DS_1$) a simpler system based on very fast handcrafted image feature extraction and support vector machine classification and ($DS_2$) a more complex system based on a deep neural network, Vgg-16, for image feature extraction and classification. $DS_1$ is much faster than $DS_2$, but it is less accurate than $DS_2$. Fortunately, the errors of $DS_1$ are not the same of $DS_2$. During training, we use a validation set to learn the probabilities of misclassification by $DS_1$ on each class based on its confidence values. When $DS_1$ quickly classifies all images from a microscopy slide, the method selects a number of images with higher chances of misclassification for characterization and reclassification by $DS_2$. Our hybrid system can improve the overall effectiveness without compromising efficiency, being suitable for the clinical routine -- a strategy that might be suitable for other real applications. As demonstrated on large datasets, the proposed system can achieve, on average, 94.9%, 87.8%, and 92.5% of Cohen's Kappa on helminth eggs, helminth larvae, and protozoa cysts, respectively., Comment: 18 pages, 11 figures

The construction of generalized Backlund transformation for the $A_n$ Affine Toda hierarchy is proposed in terms of gauge transformation acting on the zero curvature representation. Such construction is based upon the graded structure of the underlying affine algebra which induces a classification of generalized Backlund transformations. Moreover, explicit examples for $su(3)$ and $su(4)$ lead to uncover interesting composition properties of various types of Backlund transformations. The universality character of the gauge-Backlund transformation method is extended to all equations of the hierarchy. Such interesting property provides a systematic framework to construct Backlund transformations to higher flow equations. Explicit example for the simplest higher flow of the $sl(3)$ hierarchy is presented., Comment: 23 pages, Latex

We extend Painlev\'e IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlev\'e IV and II equations for special limits of the underlying parameters. We derive the underlying B\"acklund transformations, symmetry structure and requirements to satisfy Painlev\'e property., Comment: 21 pages, to appear in J. Phys. A: Math. Theor. (2020)

In this work, we revisited the ZGB model in order to study the behavior of its phase diagram when two well-known random networks play the role of the catalytic surfaces: the Random Geometric Graph and the Erd\"{o}s-R\'{e}nyi network. The connectivity and, therefore, the average number of neighbors of the nodes of these networks can vary according to their control parameters, the neighborhood radius $\alpha$ and the linking probability $p$, respectively. In addition, the catalytic reactions of the ZGB model are governed by the parameter $y$, the adsorption rate of carbon monoxide molecules on the catalytic surface. So, to study the phase diagrams of the model on both random networks, we carried out extensive steady-state Monte Carlo simulations in the space parameters ($y,\alpha$) and ($y,p$) and showed that the continuous phase transition is greatly affected by the topological features of the networks while the discontinuous one remains present in the diagram throughout the interval of study., Comment: 10 pages, 8 figures, to appear on Journal of Computational Chemistry

Mathematical and computational modeling approaches are increasingly used as quantitative tools in the analysis and forecasting of infectious disease epidemics. The growing need for realism in addressing complex public health questions is however calling for accurate models of the human contact patterns that govern the disease transmission processes. Here we present a data-driven approach to generate effective descriptions of population-level contact patterns by using highly detailed macro (census) and micro (survey) data on key socio-demographic features. We produce age-stratified contact matrices for 277 sub-national administrative regions of countries covering approximately 3.5 billion people and reflecting the high degree of cultural and societal diversity of the focus countries. We use the derived contact matrices to model the spread of airborne infectious diseases and show that sub-national heterogeneities in human mixing patterns have a marked impact on epidemic indicators such as the reproduction number and overall attack rate of epidemics of the same etiology. The contact patterns derived here are made publicly available as a modeling tool to study the impact of socio-economic differences and demographic heterogeneities across populations on the epidemiology of infectious diseases., Comment: 18 pages, 7 figures

Axelrod's model for the dissemination of culture combines two key ingredients of social dynamics: social influence, through which people become more similar when they interact, and homophily, which is the tendency of individuals to interact preferentially with similar others. In Axelrod's model, the agents are fixed to the nodes of a network and are allowed to interact with a predetermined set of peers only, resulting in the frustration of a large number of agents that end up culturally isolated. Here we modify this model by allowing the agents to move away from their cultural opposites and stay put when near their cultural likes. The comfort, i.e., the tendency of an agent to stay put in a neighborhood, is determined by the cultural similarity with its neighbors. The less the comfort, the higher the odds that the agents will move apart a fixed step size. We find that the comfort-driven mobility fragments severely the influence network for low initial cultural diversity, resulting in a network composed of only microscopic components in the thermodynamic limit. For high initial cultural diversity and intermediate values of the step size, we find that a macroscopic component coexists with the microscopic ones. The transition between these two fragmentation regimes changes from continuous to discontinuous as the step size increases. In addition, we find that for both very small and very large step sizes the influence network is severely fragmented.

We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlev\'e property. This study is informed by structure of Weyl symmetries of Painlev\'e P$_{III}$ and mixed Painlev\'e P$_{III-V}$ equations and provides insights into differences between the symmetries of Painlev\'e equations and symmetries of solvable equations on Ince's list., Comment: 9 pages, Proc. of the 32nd International Colloquium on Group Theoretical Methods in Physics (Group32), 2018, Prague, Czech Republic

The paper discusses P$_{III-V}$ equation for special values of its parameters for which this equation reduces to P$_{III}$, I$_{12}$, as well as, to some special cases of I$_{38}$ and I$_{49}$ equations from the Ince's list of $50$ second order differential equations possessing Painlev\'e property. These reductions also yield symmetries governing the reduced models obtained from the P$_{III-V}$ equation. We point out that the solvable equations on Ince's list emerge in this reduction scheme when the underlying reflections of the Weyl symmetry group no longer include an affine reflection through the hyperplane orthogonal to the highest root and therefore do not give rise to an affine Weyl group. We hypothesize that on the level of the underlying algebra and geometry this might be a fundamental feature that distinguishes the six Painlev\'e equations from the remaining $44$ solvable equations on the Ince's list., Comment: 10 pages, Proc. of the 32nd International Colloquium on Group Theoretical Methods in Physics (Group32), 2018, Prague, Czech Republic

Communication or influence networks are probably the most controllable of all factors that are known to impact on the problem-solving capability of task-forces. In the case connections are costly, it is necessary to implement a policy to allocate them to the individuals. Here we use an agent-based model to study how distinct allocation policies affect the performance of a group of agents whose task is to find the global maxima of NK fitness landscapes. Agents cooperate by broadcasting messages informing on their fitness and use this information to imitate the fittest agent in their influence neighborhoods. The larger the influence neighborhood of an agent, the more links, and hence information, the agent receives. We find that the elitist policy in which agents with above-average fitness have their influence neighborhoods amplified, whereas agents with below-average fitness have theirs deflated, is optimal for smooth landscapes, provided the group size is not too small. For rugged landscapes, however, the elitist policy can perform very poorly for certain group sizes. In addition, we find that the egalitarian policy, in which the size of the influence neighborhood is the same for all agents, is optimal for both smooth and rugged landscapes in the case of small groups. The welfarist policy, in which the actions of the elitist policy are reversed, is always suboptimal, i.e., depending on the group size it is outperformed by either the elitist or the egalitarian policies.

The motivation that the field of differential equations provide to several researchers for the challenges that have been challenging them over the decades has contributed to the strengthening of the area within mathematics. In this sense, investigating important properties of solutions of differential equations, in particular fractional, has been object of study due to the exponential growth of the fractional calculus. In this paper, we investigate the existence and uniqueness of a new class of mild and strong solution of the fractional integro-differential equations in the Hilfer fractional derivative sense in Banach space, by means of the continuously $C_{0}$-semigroup, Banach fixed point theorem and Gronwall inequality., Comment: 16 pages

The strategies adopted by individuals to select relevant information to pass on are central to understanding problem solving by groups. Here we use agent-based simulations to revisit a cooperative problem-solving scenario where the task is to find the common card in decks distributed to the group members. The agents can display only a sample of their cards and we explore different strategies to select those samples based on the confidences assigned to the cards. An agent's confidence that a particular card is the correct one is given by the number of times it observed that card in the decks of the other agents. We use a Gibbs distribution to select the card samples with the temperature measuring the strength of a noise that prevents the agents to correctly rank the cards. The group is guaranteed to find the common card in all runs solely in the infinite temperature limit, where the cards are sampled regardless of their confidences. In this case, we obtain the scaling form of the time constant that characterizes the asymptotic exponential decay of the failure probability. For finite time, however, a finite temperature yields a probability of failure that is several orders of magnitude lower than in the infinite temperature limit. The available experimental results are consistent with the decision-making model for finite temperature only.

We review the construction of the mixed Painlev\'e P$_{III-V}$ system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlev\'e P$_{III}$ or P$_{V}$. The aim of this paper is to describe solutions of P$_{III-V}$ model. In particular, we determine and classify rational, power series and transcendental solutions of P$_{III-V}$. A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P$_{III-V}$ equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order $n$ when the parameter of P$_{III-V}$ equation is quantized by integer $n \in \mathbb{Z}$., Comment: 27 pages , to appear in J. Phys. A, (2018)

Groupthink occurs when everyone in a group starts thinking alike, as when people put unlimited faith in a leader. Avoiding this phenomenon is a ubiquitous challenge to problem-solving enterprises and typical countermeasures involve the mobility of group members. Here we use an agent-based model of imitative learning to study the influence of the mobility of the agents on the time they require to find the global maxima of NK-fitness landscapes. The agents cooperate by exchanging information on their fitness and use this information to copy the fittest agent in their influence neighborhoods, which are determined by face-to-face interaction networks. The influence neighborhoods are variable since the agents perform random walks in a two-dimensional space. We find that mobility is slightly harmful for solving easy problems, i.e. problems that do not exhibit suboptimal solutions or local maxima. For difficult problems, however, mobility can prevent the imitative search being trapped in suboptimal solutions and guarantees a better performance than the independent search for any system size.

In this paper we consider the ${\cal N}=1$ supersymmetric mKdV hierarchy composed of positive odd flows embedded within an affine $\hat sl(2,1)$ algebra. Its B\"acklund transformations are constructed in terms of a gauge transformation preserving the zero curvature representation. The recursion operator relating consecutive flows is derived and shown to relate their B\"acklund transformations., Comment: 17 pages. Version accepted for publication in the Proceedings of the X International Symposium Quantum Theory and Symmetries and XII International Worshop Lie Theory and Its Applications in Physics

The integrability of the ${\cal N}=1$ supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super B\"acklund transformation. The construction of such transformation is performed by using essentially two methods: the B\"acklund-defect matrix approach and the superfield approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the B\"acklund equations in components for the first few flows of the hierarchy, namely $t_3$ and $t_5$. Secondly, the supersymmetric extension of the B\"acklund transformation in the superspace formalism is constructed for those flows. Finally, this super B\"acklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. Further integrability aspects by considering modified conserved quantities are derived from the defect matrix., Comment: 40 pages. Some comments and references added. Version accepted for publication in JHEP